Whitney’s theorem for 2-regular planar digraphs
Dan Archdeacon
Department of Mathematics and Statistics University of Vermont
Burlington, VT U.S.A.
Matt DeVos
∗Stefan Hannie Bojan Mohar
†Department of Mathematics Simon Fraser University
Burnaby, BC Canada
[email protected] [email protected] [email protected] In memory of Dan Archdeacon, our coauthor and friend.
Abstract
A digraph is 2-regular if every vertex has both indegree and outdegree two. We define an embedding of a 2-regular digraph to be a 2-cell embed- ding of the underlying graph in a closed surface with the added property that for every vertex v, the two edges directed away from v are not con- secutive in the local rotation around v. In other words, at each vertex the incident edges are oriented in-out-in-out. The goal of this article is to provide an analogue of Whitney’s theorem on planar embeddings in the setting of 2-regular digraphs. In the course of doing so, we note that Tutte’s Theorem on peripheral cycles also has a natural analogue in this setting.
1 Introduction
Although our central subject will be embeddings of certain digraphs in the sphere, we begin by introducing some general terminology for working with embedded (undi-
∗ Supported by a Canadian NSERC Discovery Grant.
† Supported in part by an NSERC Discovery Grant (Canada), by the Canada Research Chairs program, and by the Research Grant P1–0297 of ARRS (Slovenia). On leave from: IMFM, Depart- ment of Mathematics, Ljubljana, Slovenia.
rected) graphs. We will use the term surface for a closed 2-manifold without bound- ary. IfG = (V, E) is a graph, an embedding of G in a surface S is a function φ which maps V injectively to S and assigns each edge uv ∈ E to a simple curve φ(uv) in S from φ(u) to φ(v) with the added property that the curves associated to distinct edges are internally disjoint. As usual, we will identify the graphG with its image in S under φ. The faces are the connected components of the space obtained from S by deleting φ(V ) ∪ φ(E), and we have a 2-cell embedding if every face is homeomorphic to an open disc. In the case of a 2-cell embedding, each face is bounded by a closed walk inG, and we call these facial walks.
Ifφ1 and φ2 are 2-cell embeddings ofG in S1 and S2, then we say thatφ1 and φ2
are equivalent if they have the same facial walks. Throughout we will focus on 2-cell embeddings up to this equivalence, and this permits us to give a purely combinatorial description of an embedded graph by specifying the facial walks. (The embedding may be recovered from this description by treating the graph as a 1-cell complex and adding discs according to the facial walks.) The reader is referred to [9] for further details. Embeddings of digraphs have been treated previously in [1, 3, 4, 5, 6, 7, 8, 10].
D D
x
y y
x
Figure 1: Whitney flip for an undirected embedded graph
Let φ be a 2-cell embedding of a graph G in a surface S. Suppose that D ⊆ S is homeomorphic to a closed disc, and suppose that the boundary of D meets G in exactly two points, say x and y. Then there is a homeomorphism α mapping D to a closed unit disc D ⊆ R2 where α(x) and α(y) are antipodes of D. Applying the map α, followed by the mirror reflection of D through the line containing α(x) and α(y), and then applying α−1 gives us a new embedding φ0 of G in the same surface S, and we call this operation a Whitney flip of D. See Figure 1 for an example.
The following theorem shows that for a 2-connected graph embedded in the 2- sphere S2, Whitney flips allow us to move between any two embeddings.
Theorem 1.1 (Whitney [12]). Ifφ1 andφ2 are embeddings of a 2-connected graphG in the 2-sphere, then by applying a sequence of Whitney flips, φ1 can be transformed into an embedding equivalent to φ2.
For a 3-connected graphG no meaningful Whitney flip can be performed, so the above theorem immediately yields the following.
Corollary 1.2. Any two embeddings of a 3-connected graph in the 2-sphere are equivalent.
In fact, the above corollary is a key step in the proof of Whitney’s Theorem.
There is a nice proof of this corollary which comes from the work of Tutte, and we will follow his approach. For an undirected graph G we say that a cycle C ⊆ G is peripheral if C is an induced subgraph and G − V (C) is connected.
Theorem 1.3 (Tutte [11]). Every edge in a 3-connected graph is contained in at least two peripheral cycles.
Observe that for a connected graph G embedded in S2, every peripheral cycle of G must be the boundary of a face by the Jordan Curve Theorem. Based on this observation and Theorem 1.3 we have that for a 3-connected graph G embedded in S2, the facial walks are given by the peripheral cycles. Thus, Theorem 1.3 implies Corollary 1.2.
Our goal in this article is to prove theorems analogous to Theorem 1.1 and The- orem 1.3 in the setting of 2-regular digraphs.
2 Theorems of Whitney and Tutte for 2-regular digraphs
For a 2-regular digraph H, we define an embedding of H in a surface S to be an embedding of the underlying graph with the additional property that every face is bounded by a directed walk. If this holds, then for every vertexv, which is the head of two edgese1, e2 and the tail of two edgesf1, f2, the local configuration is as shown on the left in Figure 2 (i.e. the incident edges go in-out-in-out aroundv). This notion of embedding can be extended to arbitrary Eulerian digraphs and was treated earlier by several authors; see, e.g. [3, 4] or the paper [2] in this issue of the journal.
f1 v f2 e1
e2
Figure 2: Splitting a vertex
To split the vertex v, we delete v and then add back either a new edge from the tail of ei to the head of fi for i = 1, 2 or we add back a new edge from the tail of ei to the head of f3−i for i = 1, 2. As shown on the right in Figure 2, both of the new 2-regular digraphs obtained by splitting v still embed in the surface S (but the embedding need not be 2-cell any more). In general, we say that a 2-regular digraph H0 which is obtained fromH by a sequence of splits is immersed in H.
Just as any minor of an (ordinary) graph G which embeds in S also embeds in S, we have that for a 2-regular digraph H which embeds in S, every digraph immersed in H also embeds in S. In fact, there is a developing theory of 2-regular digraphs under immersion which runs parallel to the classical theory of (undirected) graphs under minors. The most significant result in this area is due to Johnson [8], who proved an analogue of the Robertson-Seymour structure theorem for 2-regular digraphs which do not immerse a given digraph H. Another theorem of Johnson is the following excluded-immersion theorem for embedding in S2. Here C3(2) is the digraph obtained from a directed cycle of length 3 by adding a second edge in parallel with every original one.
Theorem 2.1 (Johnson). A 2-regular digraph H has an embedding in S2 if and only if it does not contain C3(2) as an immersion.
D D
Figure 3: Whitney flip in a 2-regular digraph
Our goal here is to prove analogues of theorems of Whitney and Tutte in this new setting. For a 2-regular digraph H embedded in a surface S, we define a Whitney flip to be a Whitney flip of D in the underlying graph, with the added property that the two points x, y where the boundary of D meets H are both interior points of edges. This additional constraint ensures that after performing any Whitney flip, the resulting 2-regular digraph will still be embedded in S according to our definition.
Our analogue of Whitney’s Theorem is as follows.
Theorem 2.2. If φ1 and φ2 are embeddings of a connected 2-regular digraph H in the 2-sphere, then by applying a sequence of Whitney flips, φ1 can be transformed into an embedding equivalent to φ2.
Let us observe that Theorem 2.2 does not hold for embeddings of arbitrary Eule- rian digraphs. For example, take any 3-connected planar graph and replace each edge e with a pair of oppositely oriented edges e+, e− joining the same pair of vertices.
This digraph has two embeddings with directed facial walks, but the only way to come from one to the other is to exchange pairs e+ with e− for all edges at the same time. To get examples without digons, one can replace each digon with a digraph of order 7 (see Figure 3 for a clue).
As before, we have a uniqueness result for digraphs which are suitably connected.
We say that a digraphH is strongly k-edge-connected if H − F is strongly connected for every F ⊆ E(H) with |F | < k.
Corollary 2.3. Any two embeddings of a strongly 2-edge-connected 2-regular digraph in S2 are equivalent.
As was the case with Corollary 1.2, the above corollary may be proved using a suitable notion of a peripheral cycle. Consider a 2-regular digraph H embedded in S2. LetC ⊆ H be a directed cycle, and let D1, D2 be the components ofS2−C (each of which is homeomorphic to a disc). It follows from the definition of embedding of a 2-regular digraph, that for every vertexv ∈ V (C), either both edges of E(H) \ E(C) incident withv must be contained in the closure of D1, or both are contained in the closure of D2. As a consequence of this, the graph H − E(C) will be disconnected unless either D1 or D2 is a face. In light of this, we define a directed cycle C in a digraphH to be peripheral if H−E(C) is strongly connected. So, as was the case with undirected graphs, in any embedding of a 2-regular digraph in S2, every peripheral cycle must bound a face. As such, the following theorem implies Corollary 2.3.
Theorem 2.4. Every edge in a strongly 2-edge-connected Eulerian digraph is con- tained in at least two peripheral cycles.
Our proofs of Theorems 2.2 and 2.4 are natural analogues of the proofs of Theo- rems 1.1 and 1.3.
3 Proofs
We begin by proving our analogue of Tutte’s peripheral cycles theorem.
Proof of Theorem 2.4. Let e = (u, v) be an edge of H. Our first goal will be to find one peripheral cycle through e. To do this, we choose a directed path P from v to u so as to lexicographically maximize the sizes of the components of H0 = H − (E(P ) ∪ {e}). That is, we choose the path P so that the largest component of H0 is as large as possible, and subject to this the second largest is as large as possible, and so on.
We claim that H0 is connected. Suppose (for a contradiction) that H0 has com- ponentsH1, H2, . . . , Hk withk > 1 where Hk is a smallest component. LetP0 be the shortest directed path contained in P which contains all vertices in V (Hk) ∩V (P ) and suppose the start ofP0 is the vertex x and the last vertex is y. By construction, Hk must contain both x and y. Furthermore, since Hk is Eulerian, we may choose a directed path P00 inHk fromx to y. If there is a component Hi with i < k which contains a vertex in the interior of P0, then we get a contradiction to our choice of P , since we can reroute the original path along P00 instead of P0 and get a new path which improves our lexicographic ordering. Therefore, all vertices in the interior of
P0 must also be in Hk. However, in this case Hk∪ P0 is a subgraph which is sepa- rated from the rest of the graph by just two edges, and we have a contradiction to the strong 2-edge-connectivity of H. It follows that k = 1, so the cycle P ∪ {e} is indeed peripheral.
Since the cycle P ∪ {e} is peripheral, there exists a directed path Q from v to u with E(Q) ∩ E(P ) = ∅. Among all such directed paths Q we choose one so that the unique component ofH − (E(Q) ∪ {e}) which contains P is as large as possible, and subject to that we lexicographically maximize the sizes of the remaining components.
By the same argument as above, this choice will result in another peripheral cycle.
As mentioned before, the above result immediately implies Corollary 2.3 which asserts that any two embeddings of a strongly 2-edge-connected 2-regular digraph in S2 are equivalent. Next we will bootstrap this to prove our analogue of Whitney’s Theorem. For a set of verticesX ⊆ V (H) we let d+H(X) denote the number of edges with tail in X and head in V (H) \ X.
Proof of Theorem 2.2. We proceed by induction on |V (H)|. If H is strongly 2-edge- connected, then the result follows from Corollary 2.3. Otherwise, we may choose a set X ⊂ V (H) so that d+H(X) = 1 and subject to this we choose X minimal. Let u ∈ X be the vertex incident with the outgoing edge. Since H is Eulerian, there is precisely one vertex v ∈ X incident with the edge coming from V (H) \ X. Let H0 be the 2-regular digraph obtained from H by deleting all vertices not in X and then adding the edge uv. This operation may be performed to the embeddings φ1 and φ2
ofH to obtain embeddings φ01 and φ02 of H0 inS2 where the new edge follows a path fromu to v through the vertices in V (H) \ X. By the minimality of X, the digraph H0 is strongly 2-edge-connected (otherwise there would be a set Y ⊂ V (H0) = X with 1 = d+H0(Y ) ≥ d+H(Y ) contradicting the choice of X). By Corollary 2.3, the embeddings φ01 and φ02 are equivalent, and they have the same facial walks.
Next we construct a 2-regular digraph H00 from H by deleting X and adding a new edgee00joining the end vertices of the two edges between X and V (H) \ X. The embeddingsφ1 andφ2of H induce embeddings φ001 andφ002 ofH00inS2. By induction, we may choose a sequence of Whitney flips to apply toφ001 to obtain an embedding of H00 which is equivalent toφ002. The disks of the flips can be chosen in such a way that the disks never contain the new edge e00. Then the same sequence of Whitney flips may also be applied to the original embedding φ1 of H. After doing so, it follows from our analysis of H0 that the resulting embedding is either equivalent to φ2 or may be made equivalent to φ2 by performing a Whitney flip on a suitable D ⊆ S2 which contains precisely the vertices inX and intersects H in the two edges between X and V (H) \ X. This completes the proof.
Remark
This work was done during the sabbatical visit of Dan Archdeacon at Simon Fraser University from September to December 2013.
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(Received 15 Jan 2016)
